In the experimental setup (Fig. REF ), the hyper-entangled photon pairs are generated via the SPDC when a femtosecond (fs) pulsed laser passes through two 0.6-mm-thick type-I BBO (\(\beta \) -barium borate) crystals glued with orthogonally configured optic axes [1]}, [2]}, in which the polarization and OAM entanglements are achieved simultaneously in the small-angle type-I SPDC [3]}. The spatial and temporal walk-offs between photons from two BBO crystals are compensated with various YVO\(_{4}\)  crystals (see [4]} for details). Thus we directly prepare the hyper-entangled state \(\left|{\phi ^{+}} \right\rangle \otimes \left|{\nu ^{+}} \right\rangle \)  and the other fifteen hyper-entangled states can also be easily realized with the aid of two Dove prisms (DPs) and two half wave plates (HWPs) (see [4]} for details). Here the prepared OAM entanglement in the subspace has the OAMs of \(\pm \hbar \)  (i.e. \(m = 1\) ). Then the two photons enter into two input ports of a beam splitter (BS) to implement the HOM interference in two DoFs, where the arriving time of one photon is controlled by the delay line. A state measurement configuration by recording the two-photon coincidence counts in the two output port of BS is utilized to obtain the HOM interference curve between two hyper-entangled photons. For the first one \(\left|{\phi ^{+}} \right\rangle \otimes \left|{\nu ^{+}} \right\rangle \)  among the hyper-entangled states, we choose \(\left|{H} \right\rangle _{1} \left|{+1} \right\rangle _{1}\)  and \(\left|{H} \right\rangle _{2} \left|{-1} \right\rangle _{2}\)  as the two-photon measurement bases (see [4]} for all the sixteen measurement bases).
